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Properties

Label 850.2.v.d
Level $850$
Weight $2$
Character orbit 850.v
Analytic conductor $6.787$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [850,2,Mod(107,850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(850, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("850.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.v (of order \(16\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.78728417181\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 170)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 16 q^{18} - 8 q^{26} - 24 q^{27} + 8 q^{28} + 8 q^{29} - 16 q^{31} + 32 q^{33} + 8 q^{34} - 32 q^{39} - 56 q^{41} + 24 q^{42} - 16 q^{43} + 16 q^{44} + 16 q^{49} - 32 q^{51} + 16 q^{52} - 16 q^{53} - 24 q^{54} - 8 q^{56} + 120 q^{57} - 16 q^{58} + 24 q^{61} + 8 q^{62} + 24 q^{63} - 16 q^{67} + 24 q^{71} - 56 q^{72} - 88 q^{73} + 32 q^{74} - 24 q^{77} - 32 q^{78} - 104 q^{79} + 48 q^{81} - 16 q^{82} - 16 q^{83} + 96 q^{86} - 136 q^{87} - 16 q^{89} + 48 q^{91} + 8 q^{92} + 8 q^{93} - 8 q^{94} - 16 q^{97} - 72 q^{98} + 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 0.382683 − 0.923880i −3.34096 − 0.664558i −0.707107 − 0.707107i 0 −1.89250 + 2.83233i 0.616963 − 0.923351i −0.923880 + 0.382683i 7.94873 + 3.29247i 0
107.2 0.382683 − 0.923880i −1.51925 − 0.302197i −0.707107 − 0.707107i 0 −0.860584 + 1.28795i −0.134383 + 0.201119i −0.923880 + 0.382683i −0.554854 − 0.229828i 0
107.3 0.382683 − 0.923880i 0.706293 + 0.140490i −0.707107 − 0.707107i 0 0.400083 − 0.598766i −1.70679 + 2.55439i −0.923880 + 0.382683i −2.29253 − 0.949595i 0
107.4 0.382683 − 0.923880i 1.27477 + 0.253568i −0.707107 − 0.707107i 0 0.722100 − 1.08070i 0.128721 − 0.192645i −0.923880 + 0.382683i −1.21089 − 0.501569i 0
107.5 0.382683 − 0.923880i 2.87914 + 0.572697i −0.707107 − 0.707107i 0 1.63090 − 2.44082i 0.222469 − 0.332949i −0.923880 + 0.382683i 5.18983 + 2.14970i 0
143.1 0.382683 + 0.923880i −3.34096 + 0.664558i −0.707107 + 0.707107i 0 −1.89250 − 2.83233i 0.616963 + 0.923351i −0.923880 − 0.382683i 7.94873 − 3.29247i 0
143.2 0.382683 + 0.923880i −1.51925 + 0.302197i −0.707107 + 0.707107i 0 −0.860584 − 1.28795i −0.134383 − 0.201119i −0.923880 − 0.382683i −0.554854 + 0.229828i 0
143.3 0.382683 + 0.923880i 0.706293 − 0.140490i −0.707107 + 0.707107i 0 0.400083 + 0.598766i −1.70679 − 2.55439i −0.923880 − 0.382683i −2.29253 + 0.949595i 0
143.4 0.382683 + 0.923880i 1.27477 − 0.253568i −0.707107 + 0.707107i 0 0.722100 + 1.08070i 0.128721 + 0.192645i −0.923880 − 0.382683i −1.21089 + 0.501569i 0
143.5 0.382683 + 0.923880i 2.87914 − 0.572697i −0.707107 + 0.707107i 0 1.63090 + 2.44082i 0.222469 + 0.332949i −0.923880 − 0.382683i 5.18983 − 2.14970i 0
193.1 −0.923880 + 0.382683i −2.12983 − 1.42311i 0.707107 − 0.707107i 0 2.51231 + 0.499730i −0.0411526 − 0.00818577i −0.382683 + 0.923880i 1.36290 + 3.29034i 0
193.2 −0.923880 + 0.382683i −1.41027 − 0.942312i 0.707107 − 0.707107i 0 1.66353 + 0.330896i 5.02568 + 0.999670i −0.382683 + 0.923880i −0.0471409 − 0.113808i 0
193.3 −0.923880 + 0.382683i 0.432293 + 0.288849i 0.707107 − 0.707107i 0 −0.509925 − 0.101430i −3.10821 − 0.618262i −0.382683 + 0.923880i −1.04461 − 2.52190i 0
193.4 −0.923880 + 0.382683i 0.628452 + 0.419918i 0.707107 − 0.707107i 0 −0.741309 − 0.147456i −1.27154 − 0.252926i −0.382683 + 0.923880i −0.929430 − 2.24384i 0
193.5 −0.923880 + 0.382683i 2.47936 + 1.65666i 0.707107 − 0.707107i 0 −2.92460 − 0.581740i 2.11600 + 0.420899i −0.382683 + 0.923880i 2.25467 + 5.44325i 0
207.1 −0.923880 − 0.382683i −2.12983 + 1.42311i 0.707107 + 0.707107i 0 2.51231 − 0.499730i −0.0411526 + 0.00818577i −0.382683 − 0.923880i 1.36290 − 3.29034i 0
207.2 −0.923880 − 0.382683i −1.41027 + 0.942312i 0.707107 + 0.707107i 0 1.66353 − 0.330896i 5.02568 − 0.999670i −0.382683 − 0.923880i −0.0471409 + 0.113808i 0
207.3 −0.923880 − 0.382683i 0.432293 − 0.288849i 0.707107 + 0.707107i 0 −0.509925 + 0.101430i −3.10821 + 0.618262i −0.382683 − 0.923880i −1.04461 + 2.52190i 0
207.4 −0.923880 − 0.382683i 0.628452 − 0.419918i 0.707107 + 0.707107i 0 −0.741309 + 0.147456i −1.27154 + 0.252926i −0.382683 − 0.923880i −0.929430 + 2.24384i 0
207.5 −0.923880 − 0.382683i 2.47936 − 1.65666i 0.707107 + 0.707107i 0 −2.92460 + 0.581740i 2.11600 − 0.420899i −0.382683 − 0.923880i 2.25467 − 5.44325i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.r even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 850.2.v.d 40
5.b even 2 1 170.2.r.b yes 40
5.c odd 4 1 170.2.o.b ✓ 40
5.c odd 4 1 850.2.s.d 40
17.e odd 16 1 850.2.s.d 40
85.o even 16 1 170.2.r.b yes 40
85.p odd 16 1 170.2.o.b ✓ 40
85.r even 16 1 inner 850.2.v.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.o.b ✓ 40 5.c odd 4 1
170.2.o.b ✓ 40 85.p odd 16 1
170.2.r.b yes 40 5.b even 2 1
170.2.r.b yes 40 85.o even 16 1
850.2.s.d 40 5.c odd 4 1
850.2.s.d 40 17.e odd 16 1
850.2.v.d 40 1.a even 1 1 trivial
850.2.v.d 40 85.r even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 8 T_{3}^{37} - 48 T_{3}^{36} + 88 T_{3}^{35} - 72 T_{3}^{34} + 408 T_{3}^{33} + \cdots + 524288 \) acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\). Copy content Toggle raw display